Alternating Current Circuits

Alternating current (AC) circuits are made up of voltage sources and three different types of passive elements. These are resistors, inductors (i.e., small solenoids), and capacitors. Resistors satisfy Ohm's law,

$$V = I\, R,$$ (2.411)

where $R$ is the resistance, $I$ the current flowing through the resistor, and $V$ the voltage drop across the resistor (in the direction in which the current flows). (See Section 2.1.11.) Inductors satisfy

$$V = L\, \frac{dI}{dt},$$ (2.412)

where $L$ is the inductance. [See Equation (2.325).] Finally, capacitors obey

$$V = \frac{q}{C} = \left.\int_0^t I\,dt\right/C,$$ (2.413)

where $C$ is the capacitance, $q$ is the charge stored on the plate with the most positive potential, and $I=0$ for $t<0$. (See Section 2.1.13.) Note that any passive component of a real electrical circuit can always be represented as a combination of ideal resistors, inductors, and capacitors.

Let us consider the classic LCR circuit, which consists of an inductor, $L$, a capacitor, $C$, and a resistor, $R$, all connected in series with an voltage source, $V$. See Figure 2.35. The circuit equation is obtained by setting the input voltage $V$ equal to the sum of the voltage drops across the three passive elements in the circuit. Thus,

$$V = I\,R+ L\,\frac{dI}{dt} + \left.\int_0^t I\,dt\right/C.$$ (2.414)

This is an integro-differential equation which, in general, is quite difficult to solve. Suppose, however, that both the voltage and the current oscillate at some fixed angular frequency, $\omega$, so that

$$V(t) =V_0\, \exp({\rm i}\,\omega\, t),$$ (2.415)

$$I(t) = I_0 \,\exp({\rm i}\,\omega \,t),$$ (2.416)

where ${\rm i}=\!\sqrt{-1}$, and the physical solution is understood to be the real part of the previous expressions. The assumed behavior of the voltage and current is clearly relevant to electrical circuits powered by mains electricity (which oscillates at 60 hertz in the U.S. and Canada).

Figure 2.35: An LCR circuit.

Equations (2.414)–(2.416) yield

$$V_0\, \exp({\rm i}\,\omega \,t) =I_0 \, \exp({\rm i}\,\omega \,t)... ...\rm i}\,\omega \,t) +\frac{I_0 \exp({\rm i}\,\omega \,t)}{{\rm i}\,\omega\, C},$$ (2.417)

giving

$$V_0 = I_0 \left({\rm i}\,\omega\, L + \frac{1}{{\rm i}\,\omega\, C} + R\right).$$ (2.418)

It is helpful to define the impedance of the circuit:

$$Z = \frac{V}{I} = {\rm i}\,\omega\, L + \frac{1}{{\rm i }\, \omega \,C} + R.$$ (2.419)

Impedance is a generalization of the concept of resistance. In general, the impedance of an AC circuit is a complex quantity.

The average power output of the voltage source is

$$P = \langle V(t) \,I(t) \rangle,$$ (2.420)

where the average is taken over one period of the oscillation. Let us, first of all, calculate the power using real (rather than complex) voltages and currents. We can write

$$V(t) = \vert V_0\vert\, \cos(\omega\, t),$$ (2.421)

$$I(t) = \vert I_0\vert \,\cos(\omega\, t - \theta),$$ (2.422)

where $\theta$ is the phase-lag of the current with respect to the voltage. It follows that

$$P = \vert V_0\vert\, \vert I_0\vert \int_{\omega \,t = 0}^{\omega \... ...\pi} \cos(\omega\, t)\, \cos(\omega\, t - \theta)\,\,\frac{d(\omega \,t)}{2\pi}$$

$$= \vert V_0\vert\, \vert I_0\vert \int_{\omega \,t =0}^{\omega\, ... ... \cos\theta + \sin(\omega \,t) \,\sin \theta\right] \frac{d(\omega\, t)}{2\pi},$$ (2.423)

giving

$$P = \frac{1}{2}\, \vert V_0\vert\, \vert I_0\vert \cos\theta,$$ (2.424)

because $\langle \cos(\omega\, t)\,\sin(\omega\, t)\rangle = 0$ and $\langle \cos(\omega\, t) \,\cos(\omega \,t)\rangle = 1/2$. Here, $\langle\cdots\rangle\equiv \int_{\omega\,t=0}^{\omega\,t=2\pi}(\cdots)\,d(\omega\,t)/(2\pi)$. In complex representation, the voltage and the current are written

$$V(t) = \vert V_0\vert \,\exp({\rm i}\,\omega\,t),$$ (2.425)

$$I(t) = \vert I_0\vert\, \exp[{\rm i}\,(\omega \,t - \theta)].$$ (2.426)

Now,

$$\frac{1}{2} \,( V\, I^\ast +V^\ast\, I)= \vert V_0\vert\, \vert I_0\vert\, \cos\theta.$$ (2.427)

It follows from Equation (2.424) that

$$P = \frac{1}{4} \,( V\, I^\ast + V^\ast \,I) = \frac{1}{2}\, {\rm Re}(V \,I^\ast).$$ (2.428)

Making use of Equation (2.419), we find that

$$P = \frac{1}{2} \,{\rm Re}(Z)\, \vert I\vert^2 = \frac{1}{2} \frac{{\rm Re}(Z)\,\vert V\vert^2}{\vert Z\vert^2}.$$ (2.429)

Note that power dissipation is associated with the real part of the impedance. For the specific case of an LCR circuit,

$$P = \frac{1}{2} \,R \,\vert I_0\vert^{\,2}.$$ (2.430)

[See Equation (2.419).] We conclude that only the resistor dissipates energy in this circuit. The inductor and the capacitor both store energy, but they eventually return it to the circuit without dissipation.

According to Equation (2.419), the amplitude of the current that flows in an LCR circuit, for a given amplitude of the input voltage, is given by

$$\vert I_0\vert = \frac{\vert V_0\vert}{\vert Z\vert}= \frac{\vert V_0\vert}{\sqrt{(\omega \,L-1/\omega \,C)^2 + R^2}}.$$ (2.431)

As can be seen from Figure 2.36, the response of the circuit is resonant, peaking at $\omega = 1/\!\sqrt{L\,C}$, and reaching $1/\!\sqrt{2}$ of the peak value at $\omega = 1/\!\sqrt{L\,C} \pm R/(2\,L)$ (assuming that $R \ll \!\sqrt{L/C}$). For this reason, LCR circuits are used in analog radio tuners to filter out signals whose frequencies fall outside a given band.

Figure: 2.36 The characteristics of an LCR circuit. The left-hand and right-hand panes show the amplitude and phase-lag of the current versus frequency, respectively. Here, $\omega_c=1/\!\sqrt{L\,C}$ and $Z_0 = \!\sqrt{L/C}$. The solid, short-dashed, long-dashed, and dot-dashed curves correspond to $R/Z_0 = 1$, 1/2, 1/4, and 1/8, respectively.

The phase-lag of the current with respect to the voltage is given by

$$\theta = {\rm arg}(Z) = \tan^{-1}\left( \frac{\omega \,L - 1/\omega\, C } {R} \right).$$ (2.432)

[See Equation (2.419).] As can be seen from Figure 2.36, the phase-lag varies from $-90^\circ$ for frequencies significantly below the resonant frequency, to zero at the resonant frequency ( $\omega = 1/\!\sqrt{L\,C}$), to $+90^\circ$ for frequencies significantly above the resonant frequency.

It is clear that, in conventional AC circuits, the circuit equation reduces to a simple algebraic equation, and that the behavior of the circuit is summed up by the complex impedance, $Z$. The real part of $Z$ tells us the power dissipated in the circuit, the magnitude of $Z$ gives the ratio of the peak current to the peak voltage, and the argument of $Z$ gives the phase-lag of the current with respect to the voltage.